University of Surrey, Department of Mathematics, Guildford, GU2 7XH, United Kingdom

* Corresponding author: V. V. Chepyzhov

To the blessed memory of Professor V. S. Melnik

Received
September 2017Revised
March 2018Published
January 2019

Fund Project:
The research of VVC was supported by the Ministry of Education and Science of the Russian Federation (grant 14.Z50.31.0037). The work of AK and SZ was partially supported by the EPSRC grant EP/P024920/1 and the work of SZ was partially supported by the Russian Foundation for Basic Research (projects 17-01-00515 and 18-01-00524)

The paper gives a comprehensive study of Inertial Manifolds for hyperbolic relaxations of an abstract semilinear parabolic equation in a Hilbert space. A new scheme of constructing Inertial Manifolds for such type of problems is suggested and optimal spectral gap conditions which guarantee their existence are established. Moreover, the dependence of the constructed manifolds on the relaxation parameter in the case of the parabolic singular limit is also studied.